Golden Ratio Calculator
Calculate golden ratio proportions. Enter any segment to find the matching values based on φ ≈ 1.6180339887.
Enter a value and click Calculate.
How Does the Formula Work?
The golden ratio, denoted by the Greek letter phi (φ), is one of the most celebrated constants in mathematics. It equals (1 + √5) / 2, approximately 1.6180339887. Two quantities are in the golden ratio when the ratio of the larger to the smaller equals the ratio of their sum to the larger: A/B = (A+B)/A = φ. This self-similar property makes it unique among all proportions.
Inverse: 1/φ = φ − 1 ≈ 0.6180339887…
Key identity: φ² = φ + 1
Given A (longer): B = A / φ, Total = A + B = A × φ
Given B (shorter): A = B × φ, Total = A + B
Given Total: A = Total / φ, B = Total − A
Verification: A / B = (A+B) / A = φ always holds
Example: A = 100
B = 100 / 1.618… = 61.8034
Total = 100 + 61.8034 = 161.8034
Check: 100 / 61.8034 ≈ 1.618034 ✓
Mathematical Properties of φ
The golden ratio has remarkable algebraic properties. It is the only positive number whose reciprocal equals itself minus one: 1/φ = φ − 1 ≈ 0.618. It satisfies the equation φ² = φ + 1, making it a root of x² − x − 1 = 0. Its continued fraction representation is the simplest possible: 1 + 1/(1 + 1/(1 + 1/(…))), all ones. This makes φ the "most irrational" number; it is the hardest real number to approximate with simple fractions, which is why Fibonacci ratios converge to it so slowly compared to other constants like π.
The Fibonacci Connection
The Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…) has a deep connection to φ. The ratio of consecutive Fibonacci numbers approaches the golden ratio as the numbers grow: 8/5 = 1.600, 13/8 = 1.625, 21/13 ≈ 1.615, 55/34 ≈ 1.618, 89/55 ≈ 1.618. By the time you reach 987/610, the ratio matches φ to five decimal places. This calculator shows the nearest Fibonacci pair to your input, demonstrating this convergence. The exact relationship is given by Binet's formula: the nth Fibonacci number equals (φⁿ − ψⁿ) / √5, where ψ = (1 − √5)/2.
The Golden Rectangle
A golden rectangle has sides in the ratio φ : 1. Cutting a square off its long side leaves a smaller rectangle with the same ratio, because 1 / (φ − 1) = φ, so the cut can be repeated without end.
Tips & Recommendations
Enter the longer segment, shorter segment, or total; the calculator derives the other two values and verifies the φ ratio.
The golden ratio is unique: A/B = (A+B)/A = φ. The whole relates to the part exactly as the part relates to the remainder.
Every result shows the nearest Fibonacci pair. Their ratio approximates φ; the larger the numbers, the closer the match.
Use for golden rectangles, typography ratios, layout proportions, and any design where harmonious proportions matter.
Frequently Asked Questions
What is the golden ratio?
The golden ratio (φ, phi) equals (1 + √5) / 2 ≈ 1.6180339887. Two quantities are in the golden ratio when their ratio equals the ratio of their sum to the larger quantity: A/B = (A+B)/A = φ.
How does this calculator work?
Enter any one of three values: the longer segment (A), the shorter segment (B), or the total (A+B). The calculator derives the other two values using the golden ratio.
What is the connection to Fibonacci numbers?
The ratio of consecutive Fibonacci numbers approaches the golden ratio. 8/5 = 1.6, 13/8 = 1.625, 21/13 ≈ 1.615, 89/55 ≈ 1.618. The larger the numbers, the closer the ratio gets to φ.
What is the golden angle?
The golden angle is 360° × (2 − φ) = 360° / φ² ≈ 137.5°. It splits a full turn into two arcs, about 137.5° and 222.5°, whose lengths are in the golden ratio.
What is the inverse of the golden ratio?
1/φ ≈ 0.6180339887. Uniquely, 1/φ = φ − 1. This means the golden ratio is the only positive number whose reciprocal equals itself minus one.
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